As for doing a car all at once, it doesn't have to be done that way. As a disabled vet, I normally split my car in to sections; fully completing one before starting another. Horizontal surfaces can take me a day, then verticals on another. Problem areas like front and rear bumpers get extra attention. I've also found my horizontal surfaces need polishing/waxing more often; possibly due to them always receiving direct sunlight.
In your case, you could easily wash the entire car, polish the hood with a quality polish (my personal favorite is Meg's #80 Speed Glaze, but there are many other good ones out there). From the looks of the pic, it appears there's some etching going on there, so I think an abrasive polish would be needed. Keep in mind that when I say abrasive, I am referring to how it "polishes" and not necessarily its aggressiveness. Speed Glaze or even Meg's DACP (#83) would be safe for you to use. But start out with the least aggressive (Speed Glaze or similar) first. If doing by hand, Scratch X could work good for you, too (it is an abrasive type polish, but won't damage your paint). Stay away from "compounds", as they are typically too aggressive for such minor spots. And they can wreck havoc on a finish if not used by an experienced person with the right tools.
After you polish out the spots, apply your wax or sealant of choice. I personally use Meg's NXT topped with #16 for my arctic pearl white Maxima. 4 Star UPP is also a good sealant I have, and I use it just for "something different" every once in awhile. There are many other good ones out there, and I'm sure you'll get more recommendations based on people's personal recommendations. Keep in mind that none of them are "wrong", just different personal preferences.
Almost forgot. If your car hasn't been clayed, it would be worth doing it. I would clay before polishing, then polish to your heart's content. The polishing will remove anything like clay residue, so washing after claying won't be necessary.
Congrats on getting the PC. I think you'll be polishing/waxing your car more often now. And marks like you have now will no longer be a worry. In fact, they'll be an "excuse".
Welcome to the sickness!
Dave